The learning rate selection strategy presented in [1]} is motivated by the weighted likelihood bootstrap approach of [2]}, which was shown to generate bootstrap samples that have the same asymptotic distribution as Bayesian posterior distribution under a correctly specified model. For the case of a misspecified model, [1]} proposed a modified the weighted likelihood bootstrap approach which replaces the ordinary bootstrap with the Bayesian bootstrap, and establish its asymptotic limiting distribution. Then following a strategy similar to that in [4]} described above, they propose to choose \(\eta \)  in order to match the limiting \(\eta \) -generalized Bayes posterior to that of this modified likelihood bootstrap. They then show that, using the notation defined at the end of Section REF , an “oracle” learning rate is
\( \eta ^\star = \frac{\mathrm {tr}(V_{\theta ^\dagger } \Lambda _{\theta ^\dagger }^{-1} V_{\theta ^\dagger })}{\mathrm {tr}(V_{\theta ^\dagger })}. \) 
